Brownian dynamics simulations of needle chain and nugget chain polymer models--rigid constraint conditions versus infinitely stiff springs

It is well known that orientational correlations appear in polymer chain models when the subunits are linked by ball-socket joints implemented as rigid constraint conditions. These correlations do not appear when the subunits are connected by springlike potential forces, even in the limit of infinitely stiff springs. In a widely used class of algorithms for Brownian dynamics simulations, inertia effects are ignored. However, in the recently introduced needle chain and nugget chain algorithms, the rigid constraint correlations depend on the mass and moment of inertia. This inconsistency does not appear in the bead-rod (Kramers) polymer chain model, which also has orientational correlations introduced by rigid constraint conditions. Explicit expressions for the correlation functions are given for thermodynamic equilibrium states. Analytical expressions for the associated forces ("metric forces") and simulation results showing how the rigid constraint correlations influence dynamical properties, are also presented. Further we discuss the physical relevance of these correlations and show via simulations that their influence on stationary and dynamical properties depend significantly on chain length. We further show that if the metric forces are removed, algorithms designed with rigid constraint conditions describe a chain of segments connected by infinitely stiff springs. Finally we show that the results presented here for needle chains are relevant also for the bead-rod (Kramers) chain model, making it possible to simulate a bead-spring chain with infinitely stiff springs.     

Folding kinetics of a polymer

We present the results of computer simulations giving a kinetic insight into the liquid-to-solid transition of a homopolymer chain with short-range interactions. By calculating the absolute rates in each direction of the transition, using molecular dynamics employing the forward flux sampling scheme, we provide the phase diagram based on purely kinetic data, and compare it with the results from Monte Carlo simulations. Additionally, we present and discuss a remarkably simple and general relation between the polymer topology and the folding pathway, and show that the eigenvalue spectrum of a matrix defined by non-bonded contacts (the Laplacian matrix) provides an insight into the nonequilibrium ensembles of these trajectories. In particular, the Laplacian matrix seems to identify a large fraction of configurations on the folding pathway at the free energy maximum that have a very low probability of reaching the crystallized state. This implies that the eigenvalues of this matrix may be suitable additional reaction coordinates to describe the folding transition of chain molecules.     

Nonequilibrium molecular dynamics of the rheological and structural properties of linear and branched molecules. Simple shear and poiseuille flows; instabilities and slip

Nonequilibrium molecular-dynamics simulations are performed for linear and branched chain molecules to study their rheological and structural properties under simple shear and Poiseuille flows. Molecules are described by a spring-monomer model with a given intermolecular potential. The equations of motion are solved for shear and Poiseuille flows with Lees and Edward's [A. W. Lees and S. F. Edwards, J. Phys. C 5, 1921 (1972)] periodic boundary conditions. A multiple time-scale algorithm extended to nonequilibrium situations is used as the integration method, and the simulations are performed at constant temperature using Nose-Hoover [S. Nose, J. Chem. Phys. 81, 511 (1984)] dynamics. In simple shear, molecules with flow-induced ellipsoidal shape, having significant segment concentrations along the gradient and neutral directions, exhibit substantial flow resistance. Linear molecules have larger zero-shear-rate viscosity than that of branched molecules, however, this behavior reverses as the shear rate is increased. The relaxation time of the molecules is associated with segment concentrations directed along the gradient and neutral directions, and hence it depends on structure and molecular weight. The results of this study are in qualitative agreement with other simulation studies and with experimental data. The pressure (Poiseuille) flow is induced by an external force F(e) simulated by confining the molecules in the region between surfaces which have attractive forces. Conditions at the boundary strongly influence the type of the slip flow predicted. A parabolic velocity profile with apparent slip on the wall is predicted under weakly attractive wall conditions, independent of molecular structure. In the case of strongly attractive walls, a layer of adhered molecules to the wall produces an abrupt distortion of the velocity profile which leads to slip between fluid layers with magnitude that depends on the molecular structure. Finally, the molecular deformation under flow depends on the attractive force of the wall, in such a way that molecules are highly deformed in the case of strong attracting walls.     

Structure of poly(acrylic acid) in electrolyte solutions determined from simulations and viscosity measurements

In this work, the structure of poly(acrylic acid) (PAA) molecules in electrolyte solutions obtained from molecular dynamic simulations was compared with experimental data derived from dynamic light scattering (PCS), dynamic viscosity, and electrophoretic measurements. Simulations and measurements were carried out for polymer having a molecular weight of 12 kD for various ionic strengths of the supporting electrolyte (NaCl). The effect of the ionization degree of the polymer, regulated by the change in the pH of the solution in the range 4-9 units, was also studied systematically. It was predicted from theoretical simulations that, for low electrolyte concentration (10(-3) M) and pH = 9 (full nominal ionization of PAA), the molecule assumed the shape of a flexible rod having the effective length L(ef) = 21 nm, compared to the contour length L(ext) = 41 nm predicted for a fully extended polymer chain. For an electrolyte concentration of 0.15 M, it was predicted that L(ef) = 10.5 nm. For a lower ionization degree, a significant folding of the molecule was predicted, which assumed the shape of a sphere having the radius of 2 nm. These theoretical predictions were compared with PCS experimental measurements of the diffusion coefficient of the molecule, which allowed one to calculate its hydrodynamic radius R(H). It was found that R(H) varied between 6.6 nm for low ionic strength (pH = 9) and 5.8 nm for higher ionic strength (pH = 4). The R(H) values for pH = 9 were in a good agreement with theoretical predictions of particle shape, approximated by prolate spheroids, bent to various forms. On the other hand, a significant deviation from the theoretical shape predictions occurring at pH = 4 was interpreted in terms of the chain hydration effect neglected in simulations. To obtain additional shape information, the dynamic viscosity of polyelectrolyte solutions was measured using a capillary viscometer. It was found that, after considering the correction for hydration, the experimental results were in a good agreement with the Brenner's viscosity theory for prolate spheroid suspensions. The effective lengths derived from viscosity measurements using this theory were in good agreement with values predicted from the molecular dynamic simulations.     

Self-organization in protein folding and the hydrophobic interaction

Self-organization is a critical aspect of living systems. During the folding of protein molecules, the hydrophobic interaction plays an important role in the collapse of the peptide chain to a compact shape. As the hydrophobic core tightens and excludes water, not only does the number of hydrophobic side chain contacts increase, but stabilization is further enhanced by an increase in strength of each hydrophobic interaction between side chains in the core. Thus, the self-organization of the protein folding process augments itself by enhancing the stability of the core against large-scale motions that would unfold the protein. Through calculations and computer simulations on a model four-helix bundle protein, we show how the strengthening of the hydrophobic interaction is crucial for stabilizing the core long enough for completion of the folding process and quantitatively manifests self-organizing dynamical behavior.     

高分子链形状与尺寸关联的Monte Carlo模拟

运用MonteCarlo方法对线型高分子链格点模型的构型进行了模拟,研究了构型的尺寸(采用平方末端距R²,平方回转半径S²来表征)和形状(由非球形因子A表征)之间的关联.对任何长度的高分子链,其关联系数CA,R²和CA,S²均为正值,表明高分子链的形状与尺寸之间存在正关联,即尺寸小的构型其非球形因子A一般也小,反之尺寸大的构型其非球形因子A一般也大.关联系数CA,R²和CA,S²均随链长的增大而减小,近似地与链长的倒数(n^-1)成正比.研究还表明,关联系数的极限值(链长n很大时)与格点的类型无关,与链样本产生的方式也无关,但与链是否考虑排斥体积有关,考虑了排斥体积后,关联系数增大.     

Low-radii transitions in co-assembled cationic-anionic cylindrical aggregates

We investigate the formation of charged patterns on the surface of cylindrical micelles from co-assembled cationic and anionic amphiphiles. The competition between the net incompatibility chi (which arises from the different chemical nature of oppositely charged molecules) and electrostatic interactions (which prevent macroscopic segregation) results in the formation of surface domains. We employ Monte Carlo simulations to study the domains at thermal equilibrium. Our results extend previous work by studying the effect of the Bjerrum length l(B) at different values of the cylinder's radius R and chi and analyze how it affects the transition between helical, ring, and isotropic patterns. A critical surface in the space (l(B), R, chi) separating these three phases is found, and we show how it corresponds to a first-order phase transition. This confirms that the Bjerrum length l(B) is a significant parameter in the control of the helical-ring transition; the ring pattern is strongly associated with short-range forces, whereas the helical pattern develops from dominant long-range electrostatic interactions.     

端基附壁高分子模型链的塌缩转变

借MonteCarlo和模拟退火方法研究了端基附壁高分子尾形链在不同温度下的形态变化 ,链的模型采用有最近邻相互吸引作用的自避行走 .计算机实验结果表明 ,对于端基附壁的高分子尾形链 ,与自由链一样 ,当温度逐渐降低时高分子链会发生从松散的无规线团到紧密球体的塌缩转变 .计算机模拟得到了端基附壁链的均方末端距及其分量 ,均方回转半径及分布随温度的变化 .由于刚性壁的影响 ,使得有限链长的高分子尾形链与自由链相比 ,其表示链尺寸 温度关系的曲线要稍低 .模拟还发现 ,在高温时壁对链形状的影响比较大 ,壁垂直方向上尺寸明显大于平行方向的尺寸 ,后者接近于自由链的尺寸分量 .然而 ,低于θ温度时 ,尤其是完全塌缩之后 ,壁对链形状的影响已经很小 .     

Correlation of thermodynamic modeling and molecular simulations for liquid-liquid equilibrium of ternary polymer mixtures based on a phenomenological scaling method

In our previous study [S.Y. Oh, Y.C. Bae, J. Phys. Chem. B 114 (2010) 8948-8953], we presented a new method to predict liquid-liquid equilibria in ternary simple liquid mixtures by using a combination of a thermodynamic model and molecular simulations. As a continuation of that effort, we extend our previously developed method to ternary polymer systems. In the simulations, we used the dummy atoms to calculate the pair interaction energy values between the polymer segments and the solvent molecules. Furthermore, a thermodynamic model scaling concept is introduced to consider the chain length dependence of the energy parameters. This method was applied to ternary mixtures incorporating low to high molecular weight polymers. The method presented here well described the experimental observations using one or no adjustable parameters.     

Polymer chain dynamics at interfaces: role of boundary conditions at solid interface

Using classical molecular dynamics simulations, we study the dynamical properties of a single polymer chain dissolved in an explicit solvent and strongly adsorbed at solid-liquid interface. To circumvent a serious challenge posed by finite size effects due to long-range hydrodynamic effects, we developed a correction procedure that substantially limits the finite size effects. Concurrently, we provide an analysis of distinctly different size effects in the directions transverse and normal to the interface. We find that on analytically smooth interfaces, corresponding to the slip boundary condition, the motions of the polymer chain and the surrounding solvent are hydrodynamically coupled. This leads to the chain diffusion coefficient D scaling with the chain degree of polymerization N as D approximately N(-3/4), consistent with the Zimm dynamics for strongly adsorbed chains. Introduction of transverse forces at the interface results in loss of correlation between the motion of the polymer chain and the solvent. Consequently, D approximately N(-1), which is a characteristic of the Rouse dynamics.     

Thermodynamic modelling of polymer solutions with a modified Staverman equation

A model describing the thermodynamic behaviour of polymer solutions is derived which explicitly accounts for the flexibility of the polymer chains. Based on computer simulations on various lattices it is shown that the flexibility of a polymer chain can be modelled by distinguishing different polymer conformations. Here each conformation is characterized by its corresponding number of external contact sites. The equilibrium between the different conformations is then solved for any polymer concentration and any combination of interaction energies utilizing a modified Staverman equation. The model predictions are in good agreement with the results of the computer simulations which were performed using the simple-sampling and the slithering-snake algorithm. Since the knowledge of the distribution of the conformations of a single polymer chain on an empty lattice is a prerequisite to perform the model calculations, Poisson distribution functions are fitted to the results of the corresponding computer simulations. The generalization of these distribution functions not only facilitates the use of the new model but also allows to model polymers of varying chain stiffness.     

Fluctuation-dissipation relations for polymer systems. I. The molecular weight dependence of the viscosity

Generalised Langevin equations are used to describe the motion of interacting polymer molecules. In these equations the many-body aspects of the problem are incorporated into generalised friction functions and random forces. The so-called second fluctuation-dissipation relation gives a general relation between these two quantities and enables us to relate the spatial correlations present in the environment to a normal mode dependent friction coefficient and force constant. We then show how the various regimes of molecular weight dependence of the viscosity can be understood in a fairly general and non-specific way in terms of the spatial correlations of the random forces representing the rest of the system of polymer and solvent molecules. In particular a molecular weight dependence of M³ is shown to be a general feature of a spatially corelated environment.     

Elasticity of flexible and semiflexible polymers with extensible bonds in the Gibbs and Helmholtz ensembles

Stretching experiments on single molecules of arbitrary length opened the way for studying the statistical mechanics of small systems. In many cases in which the thermodynamic limit is not satisfied, different macroscopic boundary conditions, corresponding to different statistical mechanics ensembles, yield different force-displacement curves. We formulate analytical expressions and develop Monte Carlo simulations to quantitatively evaluate the difference between the Helmholtz and the Gibbs ensembles for a wide range of polymer models of biological relevance. We consider generalizations of the freely jointed chain and of the worm-like chain models with extensible bonds. In all cases we show that the convergence to the thermodynamic limit upon increasing contour length is described by a suitable power law and a specific scaling exponent, characteristic of each model.     

高分子链构象性质和溶解性的Monte Carlo模拟

运用Monte Carlo方法对线型高分子链格点模型的构象进行了模拟,研究了单链体系的构象的尺寸（采用均方末端距,均方回转半径来表征）、形状（采用平均非球形因子来表征）和溶解性随溶剂与链段间相互作用能的变化情况.结果表明,、和随着εPS的增大而减小,具有相同的变化规律;随着Δε增大,溶剂变得越来越不利于溶解,高分子链的形状蜷曲为椭球形,高分子线团相互穿插交叠.     

Towards interactive 3D graphics in chemistry publications

In this paper, we describe how it is possible to extend the visualization and publishing capabilities of the ubiquitous Adobe portable document format (PDF) from the static 2-dimensional (2D) pictures to interactive 3-dimensional (3D) models, 3D-PDF. Embedding 3D scenes of molecules and molecular properties into the electronic Computational Chemistry publications could be very important for visualization of complex molecular properties. No additional 3D-rendering software is required, because the 3D information has been already integrated within the PDF document, which can be displayed by any computer with up-to-date Adobe Acrobat software installed. We briefly describe a computer program, Jamberoo, which can be used for creating 3D-PDF documents and demonstrate several simple examples of integrated 3D models within the PDF. Finally, we show some possibilities of Acrobat JavaScript programming for creating dynamic 3D-PDF content with the elements of the Graphical User Interface (GUI).     

Monte Carlo simulation studies of polymer systems

Polymers molecules in solution or melt are more or less flexible and continuously change their shape and size. Thus, characteristic properties of the system fluctuate around statistical mean values which are dependent on the concentration of the solution, on the quality of the solvent used, and on the specific structure of the molecules, e.g. linear or star-branched. The most direct approach to these quantities on a molecular level are computer simulations. Due to restrictions of computer power fully atomistic simulations of macromolecules are presently still at the beginning but several arguments justify the use of simplified models. The most efficient way dealing with polymer systems are Monte Carlo simulations based on lattice chains, at least as long as static properties are of interest only. In the present paper a short introduction to the field is given and selected examples are presented in order to demonstrate the usefulness of these methods.     

Brownian dynamics simulations of a flexible polymer chain which includes continuous resistance and multibody hydrodynamic interactions

Using methods adapted from the simulation of suspension dynamics, we have developed a Brownian dynamics algorithm with multibody hydrodynamic interactions for simulating the dynamics of polymer molecules. The polymer molecule is modeled as a chain composed of a series of inextensible, rigid rods with constraints at each joint to ensure continuity of the chain. The linear and rotational velocities of each segment of the polymer chain are described by the slender-body theory of Batchelor [J. Fluid Mech. 44, 419 (1970)]. To include hydrodynamic interactions between the segments of the chain, the line distribution of forces on each segment is approximated by making a Legendre polynomial expansion of the disturbance velocity on the segment, where the first two terms of the expansion are retained in the calculation. Thus, the resulting linear force distribution is specified by a center of mass force, couple, and stresslet on each segment. This method for calculating the hydrodynamic interactions has been successfully used to simulate the dynamics of noncolloidal suspensions of rigid fibers [O. G. Harlen, R. R. Sundararajakumar, and D. L. Koch, J. Fluid Mech. 388, 355 (1999); J. E. Butler and E. S. G. Shaqfeh, J. Fluid Mech. 468, 204 (2002)]. The longest relaxation time and center of mass diffusivity are among the quantities calculated with the simulation technique. Comparisons are made for different levels of approximation of the hydrodynamic interactions, including multibody interactions, two-body interactions, and the "freely draining" case with no interactions. For the short polymer chains studied in this paper, the results indicate a difference in the apparent scaling of diffusivity with polymer length for the multibody versus two-body level of approximation for the hydrodynamic interactions.     

Influence of bilayer fluctuations on the steric interactions between polymer-grafted bilayers

We propose a mechanism, through which the low membrane bending rigidity changes the steric polymer repulsion between polymer-grafted bilayers. We calculate the energy of a polymer chain trapped between two thermally undulating surfaces. The proposed treatment is in terms of de Gennes' theory of fluctuations in liquid crystals, later developed by Helfrich for membranes. We show that the confinement energy of the polymer increases due to membrane fluctuations. Also, the repulsive forces between soft polymer-grafted bilayers can extend further from the surface than the height of the polymer layer.     

A single particle model to simulate the dynamics of entangled polymer melts

We present a computer simulation model of polymer melts representing each chain as one single particle. Besides the position coordinate of each particle, we introduce a parameter n(ij) for each pair of particles i and j within a specified distance from each other. These numbers, called entanglement numbers, describe the deviation of the system of ignored coordinates from its equilibrium state for the given configuration of the centers of mass of the polymers. The deviations of the entanglement numbers from their equilibrium values give rise to transient forces, which, together with the conservative forces derived from the potential of mean force, govern the displacements of the particles. We have applied our model to a melt of C(800)H(1602) chains at 450 K and have found good agreement with experiments and more detailed simulations. Properties addressed in this paper are radial distribution functions, dynamic structure factors, and linear as well as nonlinear rheological properties.     

On binding of DNA-bending proteins to DNA minicircles

We present a theoretical study of binding of DNA-bending proteins to circular DNA, using computer simulations of the wormlike chain model of DNA. We find that the binding affinity is affected by the bending elasticity and the conformational entropy of the polymer and that while protein adsorption is identical on open and closed long DNA molecules, there is significant enhancement of binding on DNA minicircles, compared to their linear counterparts. We also find that the ratio of the radii of gyration of open and closed chains depends on protein concentration for short DNA molecules. Experimental tests of our predictions are proposed.